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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1843 Excerpt:... simpler imaginary root J--1. (F) For calculating with imaginary (square) roots, there are of course no other laws or formulae but those which were given for general (square) roots, namely those summed up in (C), which however must be applied with the proper precautionary rules in (D). Namely, we shall have 7.Jl-2.Ji=5.JT, JT.Jl=(JTy-h only when we have been able so to arrange the calculations before hand that we are perfectly sure, that J--l, wherever it occurs, only represents one and the same of the two forms + J--1 and--J--1, which have in common with it its own essential property (viz. that when potentiated by 2 the result is-1). In general therefore we must write, agreeably to the formulae (C) 7.VZ22N/=T = (7_2).V=TJ and JH.JlJ+l=+l. Section 38. The general solution of the general quadratic equation ax2 + bx + c = 0, The neglect of this simple rule is, as the history of mathematical analysis teaches us, a fruitful source of contradictions, or so-called "paradoxes of calculation." now gives us without further difficulty b Jb2-4ac x =----±--. 2 a 2a It gives for the unknown expression x, two and no more than two (unequal) values (i.e. forms which when substituted for x make the equation ax3+bx + c = 0 correct (identical), viz. 0 = 0). If we now consider this general solution in the particular case when a, b, c are no longer mere supporters of the operations, but are already considered as actual numbers (i. e. as expressions in cyphers of determinate form) then the two values of x are both actual or both imaginary, according as b2--4sac is positive or negative. But in this last case we may write 2a 2a and we consequently perceive that in this case both values of x may be reduced to the form p + qj--l where p and q are actual. Section 39. F...